A049103 -id:A049103 - cp's OEIS frontend

A274357 Numbers n such that n and n+1 both have 8 divisors.

104, 135, 189, 230, 231, 285, 296, 344, 374, 375, 429, 434, 609, 645, 663, 664, 741, 776, 782, 805, 874, 902, 903, 969, 986, 1001, 1015, 1022, 1029, 1065, 1085, 1095, 1105, 1106, 1112, 1130, 1161, 1208, 1221, 1245, 1265, 1269, 1309, 1310, 1334, 1335, 1374, 1406, 1431

Offset: 1

Charles R Greathouse IV, Jun 19 2016

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A005237 and A030626.
Numbers n such that n and n+1 both have k divisors: A039832 (k=4), A049103 (k=6), A274357 (k=8), A215197 (k=10), A174456 (k=12), A274358 (k=14), A274359 (k=16), A274360 (k=18), A274361 (k=20), A274366 (k=22), A274362 (k=24), A274363 (k=26), A274364 (k=28), A274365 (k=30).
Cf. A000005.

SequencePosition[DivisorSigma[0,Range[2000]],{8,8}][[All,1]] (* Harvey P. Dale, Sep 07 2021 *)

PARI
```
is(n)=numdiv(n)==8 && numdiv(n+1)==8
```

A075036 Smaller of two smallest consecutive numbers with 2n divisors.

Original entry on oeis.org

2, 14, 44, 104, 2511, 735, 29888, 2295, 6075, 5264, 2200933376, 5984, 689278976, 156735, 180224, 21735, 2035980763136, 223244, 9399153082499072, 458864, 41680575, 701443071, 2503092614937444351, 201824, 2707370000, 29785673727, 46977524, 5475519, 1737797404898095794225152

Offset: 1

Amarnath Murthy, Sep 03 2002

nonn

There cannot be two consecutive numbers with the same odd number of divisors as both cannot be squares.
These numbers have the property that a(n) * (a(n) + 1) has 4*n^2 divisors. - David A. Corneth, Jun 24 2016
Conjecture: if a term k is even, the highest p-adic order of k (the maximum may be attained by several p's) occurs at p=2 and the highest p-adic order of k+1 occurs at p=3. If a term k is odd, the highest p-adic order of k occurs at p=3 and the highest p-adic order of k+1 occurs at p=2. - Chai Wah Wu, Mar 12 2019
a(49) = 378401464109375, a(58) = 79921490583489592950783. - Jon E. Schoenfield, May 07 2022
a(51) = 34210814718574592, a(55) = 2481402804069375, a(57) = 394311388855795712. - Jon E. Schoenfield, Nov 06 2023 - Nov 08 2023

			a(4) = 104 as tau(104) = tau(105) = 8.

Chai Wah Wu, Table of n, a(n) for n = 1..48
Ray Chandler, Known terms and upper limits.

Cf. A000005, A005237, A039832, A049103, A174456, A215197, A215199, A274357, A274358, A274359.

a[n_] := (For[k=1, ! (DivisorSigma[0, k] == 2*n && DivisorSigma[0, k+1] == 2*n), k++]; k); Array[a, 10] (* Giovanni Resta, Jun 24 2016 *)

a(n) = my(k=1); while(numdiv(k)!=2*n || numdiv(k+1)!=2*n, k++); k \\ Felix Fröhlich, Jun 24 2016

a(n) <= A215199(n-1) for n > 1. Conjecture: if p is prime, then a(p) = A215199(p-1). This conjecture is true if the conjecture in A215199 is true. The b-file of A215199 thus shows that a(p) = A215199(p-1) for prime p < 1279. - Chai Wah Wu, Mar 12 2019

a(5)-a(24) from Max Alekseyev, Mar 12 2009
a(25)-a(28) from Giovanni Resta, Jun 24 2016
a(29) from Chai Wah Wu, Mar 12 2019

A066308 a(n) = (sum of digits of n) * (product of digits of n).

Original entry on oeis.org

1, 4, 9, 16, 25, 36, 49, 64, 81, 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 0, 6, 16, 30, 48, 70, 96, 126, 160, 198, 0, 12, 30, 54, 84, 120, 162, 210, 264, 324, 0, 20, 48, 84, 128, 180, 240, 308, 384, 468, 0, 30, 70, 120, 180, 250, 330, 420, 520, 630, 0, 42, 96, 162, 240, 330

Offset: 1

Labos Elemer, Dec 13 2001

a(n) can be greater than, less than, or equal to n; see Example section.

			For n = 12, a(12) = (1 + 2)*(1*2) = 3*2 = 6 < n;
for n = 19, a(19) = (1 + 9)*(1*9) = 90 > n;
for n = 135, a(135) =(1 + 3 + 5)*(1*3*5) = 135 = n.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A007953, A007954.
Cf. A038369 (fixed points), A034710, A061672.
Cf. A049101, A049102, A049103, A049104, A049105, A049106.

asum[x_] := Apply[Plus, IntegerDigits[x]] apro[x_] := Apply[Times, IntegerDigits[x]] a[n]=asum[n]*apro[n]
sdpd[n_]:=Module[{idn=IntegerDigits[n]},Total[idn]Times@@idn]; Array[ sdpd,70] (* Harvey P. Dale, Dec 31 2011 *)

a(n) = my(d = digits(n)); vecsum(d) * vecprod(d); \\ Michel Marcus, Feb 24 2017

Edited by Jon E. Schoenfield, Jul 09 2018

A348076 Number k such that k and k+1 both have an equal number of even and odd exponents in their prime factorization (A187039).

Original entry on oeis.org

44, 75, 98, 116, 147, 171, 175, 207, 244, 332, 368, 387, 404, 507, 548, 603, 604, 656, 724, 800, 832, 844, 847, 891, 908, 931, 963, 1052, 1075, 1083, 1124, 1250, 1251, 1323, 1324, 1412, 1467, 1556, 1587, 1675, 1772, 1791, 2096, 2224, 2312, 2348, 2367, 2511, 2523

Offset: 1

Amiram Eldar, Sep 27 2021

nonn

First differs from A049103 and A074172 at n=7.

			44 is a term since 44 = 2^2 * 11 and 44 + 1 = 45 = 3^2 * 5 both have one even and one odd exponent in their prime factorization.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A187039.
A074172 is a subsequence.
Cf. A049103.

q[n_] := n == 1 || Count[(e = FactorInteger[n][[;; , 2]]), ?OddQ] == Count[e, ?EvenQ]; Select[Range[2500], q[#] && q[# + 1] &]

from sympy import factorint
def aupto(limit):
    alst, cond = [], False
    for nxtk in range(3, limit+2):
        evenodd = [0, 0]
        for e in factorint(nxtk).values():
            evenodd[e%2] += 1
        nxtcond = (evenodd[0] == evenodd[1])
        if cond and nxtcond:
            alst.append(nxtk-1)
        cond = nxtcond
    return alst
print(aupto(2523)) # Michael S. Branicky, Sep 27 2021

A049104 Numbers k such that k and k-1 both have 6 divisors.

Original entry on oeis.org

45, 76, 99, 117, 148, 172, 243, 244, 245, 333, 388, 508, 549, 604, 605, 725, 845, 909, 932, 964, 1076, 1084, 1252, 1325, 1413, 1468, 1557, 1588, 1676, 1773, 2524, 2525, 2637, 2645, 2764, 3284, 3357, 3412, 3509, 3789, 3988, 4076, 4204, 4205, 4419, 4492, 4805, 4869, 4924, 4925

Offset: 1

N. J. A. Sloane, Felice Russo, Olivier Gérard

nonn

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1986, p. 103.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Cf. A038400, A049103.

Flatten[Position[Partition[Table[DivisorSigma[0, n-1], {n, 5000}], 2, 1], ?(#=={6, 6} &)]] (* _Vincenzo Librandi, Oct 21 2012 *)

isok(n) = (numdiv(n) == 6) && (numdiv(n-1) == 6); \\ Michel Marcus, Feb 11 2016

a(n) = A049103(n)+1. - Zak Seidov, Feb 11 2016

A348345 Number k such that k and k+1 have the same positive number of noninfinitary divisors (A348341).

Original entry on oeis.org

44, 75, 98, 116, 147, 171, 242, 243, 244, 332, 387, 507, 548, 603, 604, 724, 735, 819, 844, 908, 931, 963, 1035, 1075, 1083, 1196, 1251, 1274, 1275, 1324, 1412, 1449, 1467, 1556, 1587, 1665, 1675, 1772, 1924, 1925, 1952, 1988, 2324, 2331, 2511, 2523, 2524, 2540

Offset: 1

Amiram Eldar, Oct 13 2021

nonn

First differs from A049103 at n=17.
Numbers k such that A348341(k) = A348341(k+1) > 0.
The terms are restricted to have a positive number of noninfinitary divisors, since there are many consecutive numbers without noninfinitary divisors (these are the terms of A036537).

			44 is a term since A348341(44) = A348341(45) = 2 > 0.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A036537, A049103, A348341, A348346.
Subsequence of A162643.
Similar sequences: A005237, A006049, A343819, A344312, A344313, A344314.

nid[1] = 0; nid[n_] := DivisorSigma[0, n] - Times @@ Flatten[2^DigitCount[#, 2, 1] & /@ FactorInteger[n][[;; , 2]]]; Select[Range[2500],(nid1 = nid[#]) > 0 && nid1 == nid[# + 1] &]

A348341(n) = (numdiv(n)-factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])));
isA348345(n) = { my(u=A348341(n)); (u>0&&(A348341(1+n)==u)); }; \\ Antti Karttunen, Oct 13 2021

A038400 List of pairs of consecutive numbers each with 6 divisors (duplicates removed).

Original entry on oeis.org

44, 45, 75, 76, 98, 99, 116, 117, 147, 148, 171, 172, 242, 243, 244, 245, 332, 333, 387, 388, 507, 508, 548, 549, 603, 604, 605, 724, 725, 844, 845, 908, 909, 931, 932, 963, 964, 1075, 1076, 1083, 1084, 1251, 1252, 1324, 1325, 1412, 1413, 1467, 1468, 1556, 1557, 1587, 1588

Offset: 1

Felice Russo

nonn

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, p. 103.

Harvey P. Dale, Table of n, a(n) for n = 1..3000

Cf. A049103, A049104.

SequencePosition[DivisorSigma[0,Range[1600]],{6,6}]//Flatten//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 23 2017 *)

More terms from Olivier Gérard

A356743 Numbers k such that k and k+2 both have exactly 6 divisors.

Original entry on oeis.org

18, 50, 242, 243, 423, 475, 603, 637, 722, 845, 925, 1682, 1773, 2007, 2523, 2525, 2527, 3123, 3175, 3177, 4203, 4475, 4525, 4923, 5823, 6725, 6811, 6962, 7299, 7442, 7675, 8425, 8957, 8973, 9457, 9925, 10051, 10082, 10467, 11673, 11709, 12427, 12482, 12591, 13023, 13075

Offset: 1

Jianing Song, Aug 25 2022

nonn

If an even number has exactly 6 divisors, then it is of the form 32, 4*p or 2*p^2 for an odd prime p. Note that 4*p + 2 = 2*q^2 is impossible since q^2 - 1 is divisible by 24 for prime q >= 5. As a result, if k is an even term, then it is of the form 2*p^2 such that (p^2+1)/2 is a prime (p is in A048161).

			50 is a term since 50 and 52 both have 6 divisors.

Jianing Song, Table of n, a(n) for n = 1..10000

Cf. A048161.
Numbers k such that k and k+2 both have exactly m divisors: A001359 (m=2), A356742 (m=4), this sequence (m=6), A356744 (m=8).
Cf. also A049103 (numbers k such that k and k+1 both have exactly 6 divisors).

isA356743(n) = numdiv(n)==6 && numdiv(n+2)==6

cp's OEIS Frontend

A274357 Numbers n such that n and n+1 both have 8 divisors.

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A075036 Smaller of two smallest consecutive numbers with 2n divisors.

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A066308 a(n) = (sum of digits of n) * (product of digits of n).

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A348076 Number k such that k and k+1 both have an equal number of even and odd exponents in their prime factorization (A187039).

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A049104 Numbers k such that k and k-1 both have 6 divisors.

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A348345 Number k such that k and k+1 have the same positive number of noninfinitary divisors (A348341).

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A038400 List of pairs of consecutive numbers each with 6 divisors (duplicates removed).

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A356743 Numbers k such that k and k+2 both have exactly 6 divisors.

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